3. Combining rule for the synthetic populations from multiple surveys

Qi Dong, Michael R. Elliott and Trivellore E. Raghunathan

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Assume that $Q=Q\left(Y\right)$ is the population quantity of interest depending upon the set of variables $Y$ that are collected in multiple surveys: for example, a population mean, proportion or total, a vector of regression coefficients, etc. For simplicity of exposition we assume $Q$ to be scalar. Assume that, using data from a single survey $s,$ we create $L$ synthetic populations, $S_l^{(s)},$  $l=1,\dots , L$, using the methods summarized in Section 2. Denote $Q_l^{(s)}$ as the corresponding estimate of the population quantity $Q$ obtained from synthetic population $l$ generated using data from survey $s$ (note this estimate can be obtained under a simple random sampling assumption). Dong et al. (2014) shows that, under reasonable asymptotic assumptions (sufficient sample size for the sample quantity of interest to be normally distributed, synthetic populations generated consistent with the survey design),

$$Q|S_1^{(s)},\mathrm{...},S_L^{(s)}\stackrel{·}{~}{t}_{L-1}\left({\overline{Q}}_L^{(s)},\left(1+L^{-1}\right)B_L^{(s)}\right)(3.1)$$

where ${\overline{Q}}_L^{(s)}=L^{-1}{\displaystyle {\sum }_{l=1}^L{Q}_l^{(s)}}$ is the mean of $Q$ across the $L$ synthetic populations and $B_L^{(s)}={(L-1)}^{-1}{\displaystyle {\sum }_{l=1}^L{\left(Q_l^{(s)}-{\overline{Q}}_L^{(s)}\right)}^2}$ is the between-imputation variance. The result follows immediately from Section 4.1 of Raghunathan, Reiter and Rubin (2003), and is based on the standard Rubin (1987) multiple imputation combining rules. The average “within” imputation variance is zero, since the entire population is being synthesized; hence the posterior variance of $Q$ is entirely a function of the between-imputation variance.

The combining rule obtained in (3.1) may not yield valid inference for the parameters of interest for multiple surveys, since the models to generate synthetic populations for the multiple surveys may be different. Thus, a new rule for combining estimates across multiple surveys needs to be developed.

3.1  Normal Approximation when $L$ is large

Let ${\overline{Q}}_L^{(s)}$ and $B_L^{\left(s\right)}$ be the combined estimator of the population quantity of interest and its variance for survey $s$ obtained using the combining formulas for synthetic populations ${\text{S}}_{syn}^{\left(s\right)}=\left\{{\text{S}}_l^{\left(s\right)}, l=1,\dots ,L\right\},$ $s=1,\dots ,S$ in a single survey setting. When $L$ is large, we have

$$Q|S_{syn}^{\left(1\right)},\mathrm{...},S_{syn}^{\left(S\right)}\stackrel{·}{~}N\left({\overline{Q}}_L,B_L\right) (3.2)$$

where ${\overline{Q}}_L={\displaystyle {\sum }_{s=1}^S\left({\overline{Q}}_L^{(s)}/B_L^{(s)}\right)}/{\displaystyle {\sum }_{s=1}^S\left(1/B_L^{(s)}\right)}$ and $B_L=1/{\displaystyle {\sum }_{s=1}^S\left(1/B_L^{(s)}\right)}.$ Equation (3.2) follows immediately from standard Bayesian results, assuming that 1) the true variance of ${\overline{Q}}_L^{(s)},$  $B_s,$ can be approximated by $B_L^{\left(s\right)}$ obtained from the synthetic populations as in Section 3, i.e., $\left({\overline{Q}}_L^{(s)}|Q,B_s\right)=\left({\overline{Q}}_L^{(s)}|Q,B_L^{(s)}\right)~N\left(Q,B_L^{(s)}\right)$, 2) each survey is independent, and 3) $Q$ has a non-informative prior $\pi \left(Q|B_L^{(s)}\right)\propto 1.$

3.2  T-corrected Distribution for Small/Moderate $L$

For small to moderate $L,$ the posterior distribution of $Q$ is better approximated by

$$Q|S_{syn}^{(1)},\mathrm{...},S_{syn}^{(S)}\stackrel{·}{~}{t}_{{\nu }_L}\left({\overline{Q}}_L,\left(1+L^{-1}\right)B_L\right)(3.3)$$

where ${\overline{Q}}_L$ and $B_L$ are defined as in 3.1, and degrees of freedom ${\vartheta }_L={\left(L-1\right)/{\displaystyle {\sum }_{s=1}^S\left(\left(1/b_L^{(s)}\right)/{\displaystyle {\sum }_{s=1}^S\left(1/b_L^{(s)}\right)}\right)}}^2.$ Details are available in Dong (2012), and follow the extensions of Raghuanthan et al. (2003) that were used to derive the large $L$ results.

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